I'm not aware of any airline that's designated as the official carrier for the whole league. However, many teams do have their own 'official' airlines as you've mentioned above. Many teams use charters through companies that cater to their needs (i.e. Sport Hawk). NBA has a program where teams can charter planes and a few teams have signed up for this service. Some fortunate teams own their own jets- like the Mavericks with their filthy rich Mark Cuban (remember broadcast.com?) who by the way was fined by the league- again.
And I really doubt any airline would be proud to be an official airline of a sporting league- especially the NFL. Half the players in that league are convicts/drug users/crooks... and I doubt any airline would be courting for such a bad image.
The only airline I think should be designated by the professional leagues is ConAir which is operated by the government to carry prisoners from airport to airport operating 727s. I bet a great deal of NFL players have been on that plane at least once in their lives.

The Miami HEAT use thier own 727-200 jet. Miami HEAT owner Mickey Arinson used to own Carnival Airlines/Pan Am 2. The plane was built in 1979 and used to be charterd by the HEAT until the team bought the plane in late 1999.